Is there a bigger infinity than the infinity of cardinality of the real numbers $R$ ? i.e. is there a set to which real numbers can't be mapped one-one to ?
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2Yes, the power set of $\mathbb{R}$ for example, i.e. the set of all subsets of $\mathbb{R}$, often written as $\mathcal{P}(\mathbb{R})$. – fgp Apr 07 '14 at 18:43
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1@rlartiga that is simply false. – Umberto P. Apr 07 '14 at 18:48
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The set of all real functions has this size, for instance. – A little lime Apr 08 '14 at 10:25
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Yes there is. P(R) = {every subset of R}.
There is a theorem that states that for every set g
|p(g)|>|g|.
That means that for R:
|p(R)|=2^א < א=|R|
That also means that the real numbers cannot be mapped one-one to every subset of the real numbers R
Nir Agami
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